The most commonly used spline function is the one with \( k=3 \), the so-called cubic spline function. Assume that we have in adddition to the \( n+1 \) knots a series of functions values \( y_0=f(x_0), y_1=f(x_1), \dots y_n=f(x_n) \). By definition, the polynomials \( s_{i-1} \) and \( s_i \) are thence supposed to interpolate the same point \( i \), that is $$ s_{i-1}(x_i)= y_i = s_i(x_i), $$ with \( 1 \le i \le n-1 \). In total we have \( n \) polynomials of the type $$ s_i(x)=a_{i0}+a_{i1}x+a_{i2}x^2+a_{i2}x^3, $$ yielding \( 4n \) coefficients to determine.